From an applications standpoint, the main difference between the GW distance and Sturm's distance is that the former is more amenable to computations.
Note however that \(\)  is also a non-convex problem, thus the computations only guarantee an output \(\pi \)  which is a stationary point of the functional.
There are several ways to obtain such stationary points, and all consist in updating iteratively the plan \(\pi \) .
One approach is to linearize the quadratic program as in [1]}.
It yields a sequence of OT programs where one alternates between the following two updates until convergence to a stationary point
\(\pi (x,y)&\leftarrow \int _{\times } (|d_(x,x^{\prime }) - d_(y,y^{\prime })|)\pi (x^{\prime },y^{\prime }),\\\pi &\leftarrow _{\pi \in (,)} \int _{\times } \pi \pi .\) 
